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<html xmlns="http://www.w3.org/1999/xhtml"><head><title>Owl_stats (owl.Owl_stats)</title><link rel="stylesheet" href="../../odoc.css"/><meta charset="utf-8"/><meta name="generator" content="odoc %%VERSION%%"/><meta name="viewport" content="width=device-width,initial-scale=1.0"/><script src="../../highlight.pack.js"></script><script>hljs.initHighlightingOnLoad();</script></head><body><div class="content"><header><nav><a href="../index.html">Up</a> – <a href="../index.html">owl</a> &#x00BB; Owl_stats</nav><h1>Module <code>Owl_stats</code></h1><p>Statistics: random number generators, PDF and CDF functions, and hypothesis tests. The module also includes some basic statistical functions such as mean, variance, skew, and etc.</p><nav class="toc"><ul><li><a href="#randomisation-functions">Randomisation functions</a></li><li><a href="#basic-statistical-functions">Basic statistical functions</a></li><li><a href="#mcmc:-markov-chain-monte-carlo">MCMC: Markov Chain Monte Carlo</a></li><li><a href="#hypothesis-tests">Hypothesis tests</a></li><li><a href="#discrete-random-variables">Discrete random variables</a></li><li><a href="#continuous-random-variables">Continuous random variables</a></li></ul></nav></header><section><header><h6 id="randomisation-functions"><a href="#randomisation-functions" class="anchor"></a>Randomisation functions</h6></header><dl><dt class="spec value" id="val-shuffle"><a href="#val-shuffle" class="anchor"></a><code><span class="keyword">val</span> shuffle : <span><span class="type-var">'a</span> array</span> <span>&#45;&gt;</span> <span><span class="type-var">'a</span> array</span></code></dt><dd><p>``shuffle x`` return a new array of the shuffled ``x``.</p></dd></dl><dl><dt class="spec value" id="val-choose"><a href="#val-choose" class="anchor"></a><code><span class="keyword">val</span> choose : <span><span class="type-var">'a</span> array</span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> <span><span class="type-var">'a</span> array</span></code></dt><dd><p>``choose x n`` draw ``n`` samples from ``x`` without replecement.</p></dd></dl><dl><dt class="spec value" id="val-sample"><a href="#val-sample" class="anchor"></a><code><span class="keyword">val</span> sample : <span><span class="type-var">'a</span> array</span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> <span><span class="type-var">'a</span> array</span></code></dt><dd><p>``sample x n`` draw ``n`` samples from ``x`` with replacement.</p></dd></dl></section><section><header><h6 id="basic-statistical-functions"><a href="#basic-statistical-functions" class="anchor"></a>Basic statistical functions</h6></header><dl><dt class="spec value" id="val-sum"><a href="#val-sum" class="anchor"></a><code><span class="keyword">val</span> sum : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``sum x`` returns the summation of the elements in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-mean"><a href="#val-mean" class="anchor"></a><code><span class="keyword">val</span> mean : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``mean x`` returns the mean of the elements in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-var"><a href="#val-var" class="anchor"></a><code><span class="keyword">val</span> var : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``var x`` returns the variance of elements in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-std"><a href="#val-std" class="anchor"></a><code><span class="keyword">val</span> std : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``std x`` calculates the standard deviation of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-sem"><a href="#val-sem" class="anchor"></a><code><span class="keyword">val</span> sem : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``sem x`` calculates the standard error of ``x``, also referred to as standard error of the mean.</p></dd></dl><dl><dt class="spec value" id="val-absdev"><a href="#val-absdev" class="anchor"></a><code><span class="keyword">val</span> absdev : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``absdev x`` calculates the average absolute deviation of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-skew"><a href="#val-skew" class="anchor"></a><code><span class="keyword">val</span> skew : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>?&#8288;sd:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``skew x`` calculates the skewness (the third standardized moment) of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-kurtosis"><a href="#val-kurtosis" class="anchor"></a><code><span class="keyword">val</span> kurtosis : <span>?&#8288;mean:float</span> <span>&#45;&gt;</span> <span>?&#8288;sd:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``kurtosis x`` calculates the Pearson's kurtosis of ``x``, i.e. the fourth standardized moment of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-central_moment"><a href="#val-central_moment" class="anchor"></a><code><span class="keyword">val</span> central_moment : int <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``central_moment n x`` calculates the ``n`` th central moment of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-cov"><a href="#val-cov" class="anchor"></a><code><span class="keyword">val</span> cov : <span>?&#8288;m0:float</span> <span>&#45;&gt;</span> <span>?&#8288;m1:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``cov x0 x1`` calculates the covariance of ``x0`` and ``x1``, the mean of ``x0`` and ``x1`` can be specified by ``m0`` and ``m1`` respectively.</p></dd></dl><dl><dt class="spec value" id="val-concordant"><a href="#val-concordant" class="anchor"></a><code><span class="keyword">val</span> concordant : <span><span class="type-var">'a</span> array</span> <span>&#45;&gt;</span> <span><span class="type-var">'b</span> array</span> <span>&#45;&gt;</span> int</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-discordant"><a href="#val-discordant" class="anchor"></a><code><span class="keyword">val</span> discordant : <span><span class="type-var">'a</span> array</span> <span>&#45;&gt;</span> <span><span class="type-var">'b</span> array</span> <span>&#45;&gt;</span> int</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-corrcoef"><a href="#val-corrcoef" class="anchor"></a><code><span class="keyword">val</span> corrcoef : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``corrcoef x y`` calculates the Pearson correlation of ``x`` and ``y``. Namely, ``corrcoef x y = cov(x, y) / (sigma_x * sigma_y)``.</p></dd></dl><dl><dt class="spec value" id="val-kendall_tau"><a href="#val-kendall_tau" class="anchor"></a><code><span class="keyword">val</span> kendall_tau : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``kendall_tau x y`` calculates the Kendall Tau correlation between ``x`` and ``y``.</p></dd></dl><dl><dt class="spec value" id="val-spearman_rho"><a href="#val-spearman_rho" class="anchor"></a><code><span class="keyword">val</span> spearman_rho : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``spearman_rho x y`` calculates the Spearman Rho correlation between ``x`` and ``y``.</p></dd></dl><dl><dt class="spec value" id="val-autocorrelation"><a href="#val-autocorrelation" class="anchor"></a><code><span class="keyword">val</span> autocorrelation : <span>?&#8288;lag:int</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``autocorrelation ~lag x`` calculates the autocorrelation of ``x`` with the given ``lag``.</p></dd></dl><dl><dt class="spec value" id="val-percentile"><a href="#val-percentile" class="anchor"></a><code><span class="keyword">val</span> percentile : <span>float array</span> <span>&#45;&gt;</span> float <span>&#45;&gt;</span> float</code></dt><dd><p>``percentile x p`` returns the ``p`` percentile of the data ``x``. ``p`` is between 0. and 100. ``x`` does not need to be sorted beforehand.</p></dd></dl><dl><dt class="spec value" id="val-quantile"><a href="#val-quantile" class="anchor"></a><code><span class="keyword">val</span> quantile : <span>float array</span> <span>&#45;&gt;</span> float <span>&#45;&gt;</span> float</code></dt><dd><p>``quantile x p`` returns the ``p`` quantile of the data ``x``. ``x`` does not need to be sorted beforehand. When computing several quantiles on the same data, it is more efficient to &quot;pre-apply&quot; the function, as in ``let f = quantile x in List.map f <code> 0.1 ; 0.5 </code>``, in which case the data is sorted only once.</p><dl><dt>raises Invalid_argument</dt><dd><p>if ``p`` is not between 0 and 1.</p></dd></dl></dd></dl><dl><dt class="spec value" id="val-first_quartile"><a href="#val-first_quartile" class="anchor"></a><code><span class="keyword">val</span> first_quartile : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``first_quartile x`` returns the first quartile of ``x``, i.e. 25 percentiles.</p></dd></dl><dl><dt class="spec value" id="val-third_quartile"><a href="#val-third_quartile" class="anchor"></a><code><span class="keyword">val</span> third_quartile : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``third_quartile x`` returns the third quartile of ``x``, i.e. 75 percentiles.</p></dd></dl><dl><dt class="spec value" id="val-interquartile"><a href="#val-interquartile" class="anchor"></a><code><span class="keyword">val</span> interquartile : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``interquartile x`` returns the interquartile range of ``x`` which is defined as the first quartile subtracted from the third quartile.</p></dd></dl><dl><dt class="spec value" id="val-median"><a href="#val-median" class="anchor"></a><code><span class="keyword">val</span> median : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``median x`` returns the median of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-min"><a href="#val-min" class="anchor"></a><code><span class="keyword">val</span> min : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``min x`` returns the minimum element in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-max"><a href="#val-max" class="anchor"></a><code><span class="keyword">val</span> max : <span>float array</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``max x`` returns the maximum element in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-minmax"><a href="#val-minmax" class="anchor"></a><code><span class="keyword">val</span> minmax : <span>float array</span> <span>&#45;&gt;</span> float * float</code></dt><dd><p>``minmax x`` returns both ``(minimum, maximum)`` elements in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-min_i"><a href="#val-min_i" class="anchor"></a><code><span class="keyword">val</span> min_i : <span>float array</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``min_i x`` returns the index of the minimum in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-max_i"><a href="#val-max_i" class="anchor"></a><code><span class="keyword">val</span> max_i : <span>float array</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``max_i x`` returns the index of the maximum in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-minmax_i"><a href="#val-minmax_i" class="anchor"></a><code><span class="keyword">val</span> minmax_i : <span>float array</span> <span>&#45;&gt;</span> int * int</code></dt><dd><p>``minmax_i x`` returns the indices of both minimum and maximum in ``x``.</p></dd></dl><dl><dt class="spec value" id="val-sort"><a href="#val-sort" class="anchor"></a><code><span class="keyword">val</span> sort : <span>?&#8288;inc:bool</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>``sort x`` sorts the elements in the ``x`` in increasing order if ``inc = true``, otherwise in decreasing order if ``inc=false``. By default, ``inc`` is ``true``. Note a copy is returned, the original data is not modified.</p></dd></dl><dl><dt class="spec value" id="val-argsort"><a href="#val-argsort" class="anchor"></a><code><span class="keyword">val</span> argsort : <span>?&#8288;inc:bool</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>int array</span></code></dt><dd><p>``argsort x`` sorts the elements in ``x`` and returns the indices mapping of the elements in the current array to their original position in ``x``.</p><p>The sorting is in increasing order if ``inc = true``, otherwise in decreasing order if ``inc=false``. By default, ``inc`` is ``true``.</p></dd></dl><dl><dt class="spec value" id="val-rank"><a href="#val-rank" class="anchor"></a><code><span class="keyword">val</span> rank : <span>?&#8288;ties_strategy:<span>[ `Average <span>| `Min</span> <span>| `Max</span> ]</span></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>Computes sample's ranks.</p><p>The ranking order is from the smallest one to the largest. For example ``rank <code>|54.; 74.; 55.; 86.; 56.|</code>`` returns ``<code>|1.; 4.; 2.; 5.; 3.|</code>``. Note that the ranking starts with one!</p><p>``ties_strategy`` controls which ranks are assigned to equal values:</p><ul><li>``Average`` the mean of ranks should be assigned to each value. <b>Default</b>.</li><li>``Min`` the minimum of ranks is assigned to each value.</li><li>``Max`` the maximum of ranks is assigned to each value.</li></ul></dd></dl><dl><dt class="spec type" id="type-histogram"><a href="#type-histogram" class="anchor"></a><code><span class="keyword">type</span> histogram</code><code> = <a href="../../owl-base/Owl_base_stats/index.html#type-histogram">Owl_base_stats.histogram</a></code></dt><dd><p>Type for computed histograms, with optional weighted counts and normalized counts.</p></dd></dl><dl><dt class="spec value" id="val-histogram"><a href="#val-histogram" class="anchor"></a><code><span class="keyword">val</span> histogram : <span>[ <span>`Bins of <span>float array</span></span> <span><span>| `N</span> of int</span> ]</span> <span>&#45;&gt;</span> <span>?&#8288;weights:<span>float array</span></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-histogram">histogram</a></code></dt><dd><p>``histogram bins x`` creates a histogram from values in ``x``. If bins matches `` `N n`` it will construct ``n`` equally spaced bins from the minimum to the maximum in ``x``. If bins matches `` `Bins b``, ``b`` is taken as the sorted array of boundaries of adjacent bin intervals. Bin boundaries are taken as left-inclusive, right-exclusive, except for the last bin which is also right-inclusive. Values outside the bins are dropped silently.</p><p>``histogram bins ~weights x`` creates a weighted histogram with the given ``weights`` which must match ``x`` in length. The bare counts are also provided.</p><p>Returns a histogram including the ``n+1`` bin boundaries, ``n`` counts and weighted counts if applicable, but without normalisation.</p></dd></dl><dl><dt class="spec value" id="val-histogram_sorted"><a href="#val-histogram_sorted" class="anchor"></a><code><span class="keyword">val</span> histogram_sorted : <span>[ <span>`Bins of <span>float array</span></span> <span><span>| `N</span> of int</span> ]</span> <span>&#45;&gt;</span> <span>?&#8288;weights:<span>float array</span></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-histogram">histogram</a></code></dt><dd><p>``histogram_sorted bins x`` is like ``histogram`` but assumes that ``x`` is sorted already. This increases efficiency if there are less bins than data. Undefined results if ``x`` is not in fact sorted.</p></dd></dl><dl><dt class="spec value" id="val-normalise"><a href="#val-normalise" class="anchor"></a><code><span class="keyword">val</span> normalise : <a href="index.html#type-histogram">histogram</a> <span>&#45;&gt;</span> <a href="index.html#type-histogram">histogram</a></code></dt><dd><p>``normalize hist`` calculates a probability mass function using ``hist.weighted_counts`` if present, otherwise using ``hist.counts``. The result is stored in the ``normalised_counts`` field and sums to one.</p></dd></dl><dl><dt class="spec value" id="val-normalise_density"><a href="#val-normalise_density" class="anchor"></a><code><span class="keyword">val</span> normalise_density : <a href="index.html#type-histogram">histogram</a> <span>&#45;&gt;</span> <a href="index.html#type-histogram">histogram</a></code></dt><dd><p>``normalize_density hist`` calculates a probability density function using ``hist.weighted_counts`` if present, otherwise using ``hist.counts``. The result is normalized as a density that is piecewise constant over the bin intervals. That is, the sum over density times corresponding bin width is one. If bins are infinitely wide, their density is 0 and the sum over width times density of all finite bins is the total weight in the finite bins. The result is stored in the ``density`` field.</p></dd></dl><dl><dt class="spec value" id="val-pp_hist"><a href="#val-pp_hist" class="anchor"></a><code><span class="keyword">val</span> pp_hist : Stdlib.Format.formatter <span>&#45;&gt;</span> <a href="index.html#type-histogram">histogram</a> <span>&#45;&gt;</span> unit</code></dt><dd><p>Pretty-print summary information on a histogram record</p></dd></dl><dl><dt class="spec value" id="val-ecdf"><a href="#val-ecdf" class="anchor"></a><code><span class="keyword">val</span> ecdf : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> * <span>float array</span></code></dt><dd><p>``ecdf x`` returns ``(x',f)`` which are the empirical cumulative distribution function ``f`` of ``x`` at points ``x'``. ``x'`` is just ``x`` sorted in increasing order with duplicates removed.</p></dd></dl><dl><dt class="spec value" id="val-z_score"><a href="#val-z_score" class="anchor"></a><code><span class="keyword">val</span> z_score : <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>``z_score x`` calculates the z score of a given array ``x``.</p></dd></dl><dl><dt class="spec value" id="val-t_score"><a href="#val-t_score" class="anchor"></a><code><span class="keyword">val</span> t_score : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>``t_score x`` calculates the t score of a given array ``x``.</p></dd></dl><dl><dt class="spec value" id="val-normlise_pdf"><a href="#val-normlise_pdf" class="anchor"></a><code><span class="keyword">val</span> normlise_pdf : <span>float array</span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-tukey_fences"><a href="#val-tukey_fences" class="anchor"></a><code><span class="keyword">val</span> tukey_fences : <span>?&#8288;k:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> float * float</code></dt><dd><p>``tukey_fences ?k x`` returns a tuple of the lower and upper boundaries for values that are not outliers. ``k`` defaults to the standard coefficient of ``1.5``. For first and third quartiles ``Q1`` and `Q3`, the range is computed as follows:</p><p>.. math:: (Q1 - k*(Q3-Q1), Q3 + k*(Q3-Q1))</p></dd></dl><dl><dt class="spec value" id="val-gaussian_kde"><a href="#val-gaussian_kde" class="anchor"></a><code><span class="keyword">val</span> gaussian_kde : <span>?&#8288;bandwidth:<span>[ `Silverman <span>| `Scott</span> ]</span></span> <span>&#45;&gt;</span> <span>?&#8288;n_points:int</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> * <span>float array</span></code></dt><dd><p>``gaussian_kde x`` is a Gaussian kernel density estimator. The estimation of the pdf runs in `O(sample_size * n_points)`, and returns an array tuple ``(a, b)`` where ``a`` is a uniformly spaced points from the sample range at which the density function was estimated, and ``b`` is the estimates at these points.</p><p>Bandwidth selection rules is as follows: * Silverman: use `rule-of-thumb` for choosing the bandwidth. It defaults to <code>0.9 * min(SD, IQR / 1.34) * n^-0.2</code>. * Scott: same as Silverman, but with a factor, equal to <code>1.06</code>.</p><p>The default bandwidth value is ``Scott``.</p></dd></dl></section><section><header><h6 id="mcmc:-markov-chain-monte-carlo"><a href="#mcmc:-markov-chain-monte-carlo" class="anchor"></a>MCMC: Markov Chain Monte Carlo</h6></header><dl><dt class="spec value" id="val-metropolis_hastings"><a href="#val-metropolis_hastings" class="anchor"></a><code><span class="keyword">val</span> metropolis_hastings : <span>(<span>float array</span> <span>&#45;&gt;</span> float)</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> <span><span>float array</span> array</span></code></dt><dd><p>TODO: ``metropolis_hastings f p n`` is Metropolis-Hastings MCMC algorithm. f is pdf of the p</p></dd></dl><dl><dt class="spec value" id="val-gibbs_sampling"><a href="#val-gibbs_sampling" class="anchor"></a><code><span class="keyword">val</span> gibbs_sampling : <span>(<span>float array</span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> float)</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> <span><span>float array</span> array</span></code></dt><dd><p>TODO: ``gibbs_sampling f p n`` is Gibbs sampler. f is a sampler based on the full conditional function of all variables</p></dd></dl></section><section><header><h6 id="hypothesis-tests"><a href="#hypothesis-tests" class="anchor"></a>Hypothesis tests</h6></header><dl><dt class="spec type" id="type-hypothesis"><a href="#type-hypothesis" class="anchor"></a><code><span class="keyword">type</span> hypothesis</code><code> = </code><code>{</code><table class="record"><tr id="type-hypothesis.reject" class="anchored"><td class="def field"><a href="#type-hypothesis.reject" class="anchor"></a><code>reject : bool;</code></td></tr><tr id="type-hypothesis.p_value" class="anchored"><td class="def field"><a href="#type-hypothesis.p_value" class="anchor"></a><code>p_value : float;</code></td></tr><tr id="type-hypothesis.score" class="anchored"><td class="def field"><a href="#type-hypothesis.score" class="anchor"></a><code>score : float;</code></td></tr></table><code>}</code></dt><dd><p>Record type contains the result of a hypothesis test.</p></dd></dl><dl><dt class="spec type" id="type-tail"><a href="#type-tail" class="anchor"></a><code><span class="keyword">type</span> tail</code><code> = </code><table class="variant"><tr id="type-tail.BothSide" class="anchored"><td class="def constructor"><a href="#type-tail.BothSide" class="anchor"></a><code>| </code><code><span class="constructor">BothSide</span></code></td></tr><tr id="type-tail.RightSide" class="anchored"><td class="def constructor"><a href="#type-tail.RightSide" class="anchor"></a><code>| </code><code><span class="constructor">RightSide</span></code></td></tr><tr id="type-tail.LeftSide" class="anchored"><td class="def constructor"><a href="#type-tail.LeftSide" class="anchor"></a><code>| </code><code><span class="constructor">LeftSide</span></code></td><td class="doc"><p>Types of alternative hypothesis tests: one-side, left-side, or right-side.</p></td></tr></table></dt></dl><dl><dt class="spec value" id="val-pp_hypothesis"><a href="#val-pp_hypothesis" class="anchor"></a><code><span class="keyword">val</span> pp_hypothesis : Stdlib.Format.formatter <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a> <span>&#45;&gt;</span> unit</code></dt><dd><p>Pretty printer of hypothesis type</p></dd></dl><dl><dt class="spec value" id="val-z_test"><a href="#val-z_test" class="anchor"></a><code><span class="keyword">val</span> z_test : <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``z_test ~mu ~sigma ~alpha ~side x`` returns a test decision for the null hypothesis that the data ``x`` comes from a normal distribution with mean ``mu`` and a standard deviation ``sigma``, using the z-test of ``alpha`` significance level. The alternative hypothesis is that the mean is not ``mu``.</p><p>The result ``(h,p,z)`` : ``h`` is ``true`` if the test rejects the null hypothesis at the ``alpha`` significance level, and ``false`` otherwise. ``p`` is the p-value and ``z`` is the z-score.</p></dd></dl><dl><dt class="spec value" id="val-t_test"><a href="#val-t_test" class="anchor"></a><code><span class="keyword">val</span> t_test : <span>mu:float</span> <span>&#45;&gt;</span> <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``t_test ~mu ~alpha ~side x`` returns a test decision of one-sample t-test which is a parametric test of the location parameter when the population standard deviation is unknown. ``mu`` is population mean, ``alpha`` is the significance level.</p></dd></dl><dl><dt class="spec value" id="val-t_test_paired"><a href="#val-t_test_paired" class="anchor"></a><code><span class="keyword">val</span> t_test_paired : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``t_test_paired ~alpha ~side x y`` returns a test decision for the null hypothesis that the data in ``x – y`` comes from a normal distribution with mean equal to zero and unknown variance, using the paired-sample t-test.</p></dd></dl><dl><dt class="spec value" id="val-t_test_unpaired"><a href="#val-t_test_unpaired" class="anchor"></a><code><span class="keyword">val</span> t_test_unpaired : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>?&#8288;equal_var:bool</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``t_test_unpaired ~alpha ~side ~equal_var x y`` returns a test decision for the null hypothesis that the data in vectors ``x`` and ``y`` comes from independent random samples from normal distributions with equal means and equal but unknown variances, using the two-sample t-test. The alternative hypothesis is that the data in ``x`` and ``y`` comes from populations with unequal means.</p><p>``equal_var`` indicates whether two samples have the same variance. If the two variances are not the same, the test is referred to as Welche's t-test.</p></dd></dl><dl><dt class="spec value" id="val-ks_test"><a href="#val-ks_test" class="anchor"></a><code><span class="keyword">val</span> ks_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>(float <span>&#45;&gt;</span> float)</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``ks_test ~alpha x f`` returns a test decision for the null hypothesis that the data in vector ``x`` comes from independent random samples of the distribution with CDF f. The alternative hypothesis is that the data in ``x`` comes from a different distribution.</p><p>The result ``(h,p,d)`` : ``h`` is ``true`` if the test rejects the null hypothesis at the ``alpha`` significance level, and ``false`` otherwise. ``p`` is the p-value and ``d`` is the Kolmogorov-Smirnov test statistic.</p></dd></dl><dl><dt class="spec value" id="val-ks2_test"><a href="#val-ks2_test" class="anchor"></a><code><span class="keyword">val</span> ks2_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``ks2_test ~alpha x y`` returns a test decision for the null hypothesis that the data in vectors ``x`` and ``y`` come from independent random samples of the same distribution. The alternative hypothesis is that the data in ``x`` and ``y`` are sampled from different distributions.</p><p>The result ``(h,p,d)``: ``h`` is ``true`` if the test rejects the null hypothesis at the ``alpha`` significance level, and ``false`` otherwise. ``p`` is the p-value and ``d`` is the Kolmogorov-Smirnov test statistic.</p></dd></dl><dl><dt class="spec value" id="val-var_test"><a href="#val-var_test" class="anchor"></a><code><span class="keyword">val</span> var_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>variance:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``var_test ~alpha ~side ~variance x`` returns a test decision for the null hypothesis that the data in ``x`` comes from a normal distribution with input ``variance``, using the chi-square variance test. The alternative hypothesis is that ``x`` comes from a normal distribution with a different variance.</p></dd></dl><dl><dt class="spec value" id="val-jb_test"><a href="#val-jb_test" class="anchor"></a><code><span class="keyword">val</span> jb_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``jb_test ~alpha x`` returns a test decision for the null hypothesis that the data ``x`` comes from a normal distribution with an unknown mean and variance, using the Jarque-Bera test.</p></dd></dl><dl><dt class="spec value" id="val-fisher_test"><a href="#val-fisher_test" class="anchor"></a><code><span class="keyword">val</span> fisher_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> int <span>&#45;&gt;</span> int <span>&#45;&gt;</span> int <span>&#45;&gt;</span> int <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``fisher_test ~alpha ~side a b c d`` fisher's exact test for contingency table | ``a``, ``b`` | | ``c``, ``d`` |</p><p>The result ``(h,p,z)`` : ``h`` is ``true`` if the test rejects the null hypothesis at the ``alpha`` significance level, and ``false`` otherwise. ``p`` is the p-value and ``z`` is prior odds ratio.</p></dd></dl><dl><dt class="spec value" id="val-runs_test"><a href="#val-runs_test" class="anchor"></a><code><span class="keyword">val</span> runs_test : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>?&#8288;v:float</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``runs_test ~alpha ~v x`` returns a test decision for the null hypothesis that the data ``x`` comes in random order, against the alternative that they do not, by runnign Wald–Wolfowitz runs test. The test is based on the number of runs of consecutive values above or below the mean of ``x``. ``~v`` is the reference value, the default value is the median of ``x``.</p></dd></dl><dl><dt class="spec value" id="val-mannwhitneyu"><a href="#val-mannwhitneyu" class="anchor"></a><code><span class="keyword">val</span> mannwhitneyu : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>``mannwhitneyu ~alpha ~side x y`` Computes the Mann-Whitney rank test on samples x and y. If length of each sample less than 10 and no ties, then using exact test (see paper Ying Kuen Cheung and Jerome H. Klotz (1997) The Mann Whitney Wilcoxon distribution using linked list Statistica Sinica 7 805-813), else usning asymptotic normal distribution.</p></dd></dl><dl><dt class="spec value" id="val-wilcoxon"><a href="#val-wilcoxon" class="anchor"></a><code><span class="keyword">val</span> wilcoxon : <span>?&#8288;alpha:float</span> <span>&#45;&gt;</span> <span>?&#8288;side:<a href="index.html#type-tail">tail</a></span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <span>float array</span> <span>&#45;&gt;</span> <a href="index.html#type-hypothesis">hypothesis</a></code></dt><dd><p>TODO</p></dd></dl></section><section><header><h6 id="discrete-random-variables"><a href="#discrete-random-variables" class="anchor"></a>Discrete random variables</h6></header><aside><p>The ``_rvs`` functions generate random numbers according to the specified distribution. ``_pdf`` are &quot;density&quot; functions that return the probability of the element specified by the arguments, while ``_cdf`` functions are cumulative distribution functions that return the probability of all elements less than or equal to the chosen element, and ``_sf`` functions are survival functions returning one minus the corresponding CDF function. `log` versions of functions return the result for the natural logarithm of a chosen element.</p></aside><dl><dt class="spec value" id="val-uniform_int_rvs"><a href="#val-uniform_int_rvs" class="anchor"></a><code><span class="keyword">val</span> uniform_int_rvs : <span>a:int</span> <span>&#45;&gt;</span> <span>b:int</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``uniform_rvs ~a ~b`` returns a random uniformly distributed integer between ``a`` and ``b``, inclusive.</p></dd></dl><dl><dt class="spec value" id="val-binomial_rvs"><a href="#val-binomial_rvs" class="anchor"></a><code><span class="keyword">val</span> binomial_rvs : <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``binomial_rvs p n`` returns a random integer representing the number of successes in ``n`` trials with probability of success ``p`` on each trial.</p></dd></dl><dl><dt class="spec value" id="val-binomial_pdf"><a href="#val-binomial_pdf" class="anchor"></a><code><span class="keyword">val</span> binomial_pdf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_pdf k ~p ~n`` returns the binomially distributed probability of ``k`` successes in ``n`` trials with probability ``p`` of success on each trial.</p></dd></dl><dl><dt class="spec value" id="val-binomial_logpdf"><a href="#val-binomial_logpdf" class="anchor"></a><code><span class="keyword">val</span> binomial_logpdf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_logpdf k ~p ~n`` returns the log-binomially distributed probability of ``k`` successes in ``n`` trials with probability ``p`` of success on each trial.</p></dd></dl><dl><dt class="spec value" id="val-binomial_cdf"><a href="#val-binomial_cdf" class="anchor"></a><code><span class="keyword">val</span> binomial_cdf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_cdf k ~p ~n`` returns the binomially distributed cumulative probability of less than or equal to ``k`` successes in ``n`` trials, with probability ``p`` on each trial.</p></dd></dl><dl><dt class="spec value" id="val-binomial_logcdf"><a href="#val-binomial_logcdf" class="anchor"></a><code><span class="keyword">val</span> binomial_logcdf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_logcdf k ~p ~n`` returns the log-binomially distributed cumulative probability of less than or equal to ``k`` successes in ``n`` trials, with probability ``p`` on each trial.</p></dd></dl><dl><dt class="spec value" id="val-binomial_sf"><a href="#val-binomial_sf" class="anchor"></a><code><span class="keyword">val</span> binomial_sf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_sf k ~p ~n`` is the binomial survival function, i.e. ``1 - (binomial_cdf k ~p ~n)``.</p></dd></dl><dl><dt class="spec value" id="val-binomial_logsf"><a href="#val-binomial_logsf" class="anchor"></a><code><span class="keyword">val</span> binomial_logsf : int <span>&#45;&gt;</span> <span>p:float</span> <span>&#45;&gt;</span> <span>n:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``binomial_loggf k ~p ~n`` is the logbinomial survival function, i.e. ``1 - (binomial_logcdf k ~p ~n)``.</p></dd></dl><dl><dt class="spec value" id="val-hypergeometric_rvs"><a href="#val-hypergeometric_rvs" class="anchor"></a><code><span class="keyword">val</span> hypergeometric_rvs : <span>good:int</span> <span>&#45;&gt;</span> <span>bad:int</span> <span>&#45;&gt;</span> <span>sample:int</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``hypergeometric_rvs ~good ~bad ~sample`` returns a random hypergeometrically distributed integer representing the number of successes in a sample (without replacement) of size ``~sample`` from a population with ``~good`` successful elements and ``~bad`` unsuccessful elements.</p></dd></dl><dl><dt class="spec value" id="val-hypergeometric_pdf"><a href="#val-hypergeometric_pdf" class="anchor"></a><code><span class="keyword">val</span> hypergeometric_pdf : int <span>&#45;&gt;</span> <span>good:int</span> <span>&#45;&gt;</span> <span>bad:int</span> <span>&#45;&gt;</span> <span>sample:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``hypergeometric_pdf k ~good ~bad ~sample`` returns the hypergeometrically distributed probability of ``k`` successes in a sample (without replacement) of ``~sample`` elements from a population containing ``~good`` successful elements and ``~bad`` unsuccessful ones.</p></dd></dl><dl><dt class="spec value" id="val-hypergeometric_logpdf"><a href="#val-hypergeometric_logpdf" class="anchor"></a><code><span class="keyword">val</span> hypergeometric_logpdf : int <span>&#45;&gt;</span> <span>good:int</span> <span>&#45;&gt;</span> <span>bad:int</span> <span>&#45;&gt;</span> <span>sample:int</span> <span>&#45;&gt;</span> float</code></dt><dd><p>``hypergeometric_logpdf k ~good ~bad ~sample`` returns a value equivalent to a log-transformed result from ``hypergeometric_pdf``.</p></dd></dl><dl><dt class="spec value" id="val-multinomial_rvs"><a href="#val-multinomial_rvs" class="anchor"></a><code><span class="keyword">val</span> multinomial_rvs : int <span>&#45;&gt;</span> <span>p:<span>float array</span></span> <span>&#45;&gt;</span> <span>int array</span></code></dt><dd><p>``multinomial_rvs n ~p`` generates random numbers of multinomial distribution from ``n`` trials. The probabilty mass function is as follows.</p><p>.. math:: P(x) = \frac<code>n!</code><code>{x_1</code>! \cdot\cdot\cdot <code>x_k</code>!</p><p>}</p><p>p_</p><h2 id=""><a href="#" class="anchor"></a></h2><p>^<code>x_1</code> \cdot\cdot\cdot p_<code>k</code>^<code>x_k</code></p><p>``p`` is the probabilty mass of ``k`` categories. The elements in ``p`` should all be positive (result is undefined if there are negative values), but they don't need to sum to 1: the result is the same whether or not ``p`` is normalized. For implementation details, refer to :cite:`davis1993computer`.</p></dd></dl><dl><dt class="spec value" id="val-multinomial_pdf"><a href="#val-multinomial_pdf" class="anchor"></a><code><span class="keyword">val</span> multinomial_pdf : <span>int array</span> <span>&#45;&gt;</span> <span>p:<span>float array</span></span> <span>&#45;&gt;</span> float</code></dt><dd><p>``multinomial_rvs x ~p`` return the probabilty of ``x`` given the probabilty mass of a multinomial distribution.</p></dd></dl><dl><dt class="spec value" id="val-multinomial_logpdf"><a href="#val-multinomial_logpdf" class="anchor"></a><code><span class="keyword">val</span> multinomial_logpdf : <span>int array</span> <span>&#45;&gt;</span> <span>p:<span>float array</span></span> <span>&#45;&gt;</span> float</code></dt><dd><p>``multinomial_rvs x ~p`` returns the logarithm probabilty of ``x`` given the probabilty mass of a multinomial distribution.</p></dd></dl><dl><dt class="spec value" id="val-categorical_rvs"><a href="#val-categorical_rvs" class="anchor"></a><code><span class="keyword">val</span> categorical_rvs : <span>float array</span> <span>&#45;&gt;</span> int</code></dt><dd><p>``categorical_rvs p`` returns the value of a random variable which follows the categorical distribution. This is equavalent to only one trial from ``multinomial_rvs`` function, so it is just a simple wrapping.</p></dd></dl></section><section><header><h6 id="continuous-random-variables"><a href="#continuous-random-variables" class="anchor"></a>Continuous random variables</h6></header><dl><dt class="spec value" id="val-std_uniform_rvs"><a href="#val-std_uniform_rvs" class="anchor"></a><code><span class="keyword">val</span> std_uniform_rvs : unit <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_rvs"><a href="#val-uniform_rvs" class="anchor"></a><code><span class="keyword">val</span> uniform_rvs : <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_pdf"><a href="#val-uniform_pdf" class="anchor"></a><code><span class="keyword">val</span> uniform_pdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_logpdf"><a href="#val-uniform_logpdf" class="anchor"></a><code><span class="keyword">val</span> uniform_logpdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_cdf"><a href="#val-uniform_cdf" class="anchor"></a><code><span class="keyword">val</span> uniform_cdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_logcdf"><a href="#val-uniform_logcdf" class="anchor"></a><code><span class="keyword">val</span> uniform_logcdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_ppf"><a href="#val-uniform_ppf" class="anchor"></a><code><span class="keyword">val</span> uniform_ppf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_sf"><a href="#val-uniform_sf" class="anchor"></a><code><span class="keyword">val</span> uniform_sf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_logsf"><a href="#val-uniform_logsf" class="anchor"></a><code><span class="keyword">val</span> uniform_logsf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-uniform_isf"><a href="#val-uniform_isf" class="anchor"></a><code><span class="keyword">val</span> uniform_isf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_rvs"><a href="#val-exponential_rvs" class="anchor"></a><code><span class="keyword">val</span> exponential_rvs : <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_pdf"><a href="#val-exponential_pdf" class="anchor"></a><code><span class="keyword">val</span> exponential_pdf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_logpdf"><a href="#val-exponential_logpdf" class="anchor"></a><code><span class="keyword">val</span> exponential_logpdf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_cdf"><a href="#val-exponential_cdf" class="anchor"></a><code><span class="keyword">val</span> exponential_cdf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_logcdf"><a href="#val-exponential_logcdf" class="anchor"></a><code><span class="keyword">val</span> exponential_logcdf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_ppf"><a href="#val-exponential_ppf" class="anchor"></a><code><span class="keyword">val</span> exponential_ppf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_sf"><a href="#val-exponential_sf" class="anchor"></a><code><span class="keyword">val</span> exponential_sf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_logsf"><a href="#val-exponential_logsf" class="anchor"></a><code><span class="keyword">val</span> exponential_logsf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponential_isf"><a href="#val-exponential_isf" class="anchor"></a><code><span class="keyword">val</span> exponential_isf : float <span>&#45;&gt;</span> <span>lambda:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_rvs"><a href="#val-exponpow_rvs" class="anchor"></a><code><span class="keyword">val</span> exponpow_rvs : <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>.. math:: p(x) dx = (1/(2 a Gamma(1+1/b))) * exp(-|x/a|^b) dx</p></dd></dl><dl><dt class="spec value" id="val-exponpow_pdf"><a href="#val-exponpow_pdf" class="anchor"></a><code><span class="keyword">val</span> exponpow_pdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_logpdf"><a href="#val-exponpow_logpdf" class="anchor"></a><code><span class="keyword">val</span> exponpow_logpdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_cdf"><a href="#val-exponpow_cdf" class="anchor"></a><code><span class="keyword">val</span> exponpow_cdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_logcdf"><a href="#val-exponpow_logcdf" class="anchor"></a><code><span class="keyword">val</span> exponpow_logcdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_sf"><a href="#val-exponpow_sf" class="anchor"></a><code><span class="keyword">val</span> exponpow_sf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-exponpow_logsf"><a href="#val-exponpow_logsf" class="anchor"></a><code><span class="keyword">val</span> exponpow_logsf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_rvs"><a href="#val-gaussian_rvs" class="anchor"></a><code><span class="keyword">val</span> gaussian_rvs : <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_pdf"><a href="#val-gaussian_pdf" class="anchor"></a><code><span class="keyword">val</span> gaussian_pdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_logpdf"><a href="#val-gaussian_logpdf" class="anchor"></a><code><span class="keyword">val</span> gaussian_logpdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_cdf"><a href="#val-gaussian_cdf" class="anchor"></a><code><span class="keyword">val</span> gaussian_cdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_logcdf"><a href="#val-gaussian_logcdf" class="anchor"></a><code><span class="keyword">val</span> gaussian_logcdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_ppf"><a href="#val-gaussian_ppf" class="anchor"></a><code><span class="keyword">val</span> gaussian_ppf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_sf"><a href="#val-gaussian_sf" class="anchor"></a><code><span class="keyword">val</span> gaussian_sf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_logsf"><a href="#val-gaussian_logsf" class="anchor"></a><code><span class="keyword">val</span> gaussian_logsf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gaussian_isf"><a href="#val-gaussian_isf" class="anchor"></a><code><span class="keyword">val</span> gaussian_isf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_rvs"><a href="#val-gamma_rvs" class="anchor"></a><code><span class="keyword">val</span> gamma_rvs : <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_pdf"><a href="#val-gamma_pdf" class="anchor"></a><code><span class="keyword">val</span> gamma_pdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_logpdf"><a href="#val-gamma_logpdf" class="anchor"></a><code><span class="keyword">val</span> gamma_logpdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_cdf"><a href="#val-gamma_cdf" class="anchor"></a><code><span class="keyword">val</span> gamma_cdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_logcdf"><a href="#val-gamma_logcdf" class="anchor"></a><code><span class="keyword">val</span> gamma_logcdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_ppf"><a href="#val-gamma_ppf" class="anchor"></a><code><span class="keyword">val</span> gamma_ppf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_sf"><a href="#val-gamma_sf" class="anchor"></a><code><span class="keyword">val</span> gamma_sf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_logsf"><a href="#val-gamma_logsf" class="anchor"></a><code><span class="keyword">val</span> gamma_logsf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gamma_isf"><a href="#val-gamma_isf" class="anchor"></a><code><span class="keyword">val</span> gamma_isf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_rvs"><a href="#val-beta_rvs" class="anchor"></a><code><span class="keyword">val</span> beta_rvs : <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_pdf"><a href="#val-beta_pdf" class="anchor"></a><code><span class="keyword">val</span> beta_pdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_logpdf"><a href="#val-beta_logpdf" class="anchor"></a><code><span class="keyword">val</span> beta_logpdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_cdf"><a href="#val-beta_cdf" class="anchor"></a><code><span class="keyword">val</span> beta_cdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_logcdf"><a href="#val-beta_logcdf" class="anchor"></a><code><span class="keyword">val</span> beta_logcdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_ppf"><a href="#val-beta_ppf" class="anchor"></a><code><span class="keyword">val</span> beta_ppf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_sf"><a href="#val-beta_sf" class="anchor"></a><code><span class="keyword">val</span> beta_sf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_logsf"><a href="#val-beta_logsf" class="anchor"></a><code><span class="keyword">val</span> beta_logsf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-beta_isf"><a href="#val-beta_isf" class="anchor"></a><code><span class="keyword">val</span> beta_isf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_rvs"><a href="#val-chi2_rvs" class="anchor"></a><code><span class="keyword">val</span> chi2_rvs : <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_pdf"><a href="#val-chi2_pdf" class="anchor"></a><code><span class="keyword">val</span> chi2_pdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_logpdf"><a href="#val-chi2_logpdf" class="anchor"></a><code><span class="keyword">val</span> chi2_logpdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_cdf"><a href="#val-chi2_cdf" class="anchor"></a><code><span class="keyword">val</span> chi2_cdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_logcdf"><a href="#val-chi2_logcdf" class="anchor"></a><code><span class="keyword">val</span> chi2_logcdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_ppf"><a href="#val-chi2_ppf" class="anchor"></a><code><span class="keyword">val</span> chi2_ppf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_sf"><a href="#val-chi2_sf" class="anchor"></a><code><span class="keyword">val</span> chi2_sf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_logsf"><a href="#val-chi2_logsf" class="anchor"></a><code><span class="keyword">val</span> chi2_logsf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-chi2_isf"><a href="#val-chi2_isf" class="anchor"></a><code><span class="keyword">val</span> chi2_isf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_rvs"><a href="#val-f_rvs" class="anchor"></a><code><span class="keyword">val</span> f_rvs : <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_pdf"><a href="#val-f_pdf" class="anchor"></a><code><span class="keyword">val</span> f_pdf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_logpdf"><a href="#val-f_logpdf" class="anchor"></a><code><span class="keyword">val</span> f_logpdf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_cdf"><a href="#val-f_cdf" class="anchor"></a><code><span class="keyword">val</span> f_cdf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_logcdf"><a href="#val-f_logcdf" class="anchor"></a><code><span class="keyword">val</span> f_logcdf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_ppf"><a href="#val-f_ppf" class="anchor"></a><code><span class="keyword">val</span> f_ppf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_sf"><a href="#val-f_sf" class="anchor"></a><code><span class="keyword">val</span> f_sf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_logsf"><a href="#val-f_logsf" class="anchor"></a><code><span class="keyword">val</span> f_logsf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-f_isf"><a href="#val-f_isf" class="anchor"></a><code><span class="keyword">val</span> f_isf : float <span>&#45;&gt;</span> <span>dfnum:float</span> <span>&#45;&gt;</span> <span>dfden:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_rvs"><a href="#val-cauchy_rvs" class="anchor"></a><code><span class="keyword">val</span> cauchy_rvs : <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_pdf"><a href="#val-cauchy_pdf" class="anchor"></a><code><span class="keyword">val</span> cauchy_pdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_logpdf"><a href="#val-cauchy_logpdf" class="anchor"></a><code><span class="keyword">val</span> cauchy_logpdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_cdf"><a href="#val-cauchy_cdf" class="anchor"></a><code><span class="keyword">val</span> cauchy_cdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_logcdf"><a href="#val-cauchy_logcdf" class="anchor"></a><code><span class="keyword">val</span> cauchy_logcdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_ppf"><a href="#val-cauchy_ppf" class="anchor"></a><code><span class="keyword">val</span> cauchy_ppf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_sf"><a href="#val-cauchy_sf" class="anchor"></a><code><span class="keyword">val</span> cauchy_sf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_logsf"><a href="#val-cauchy_logsf" class="anchor"></a><code><span class="keyword">val</span> cauchy_logsf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-cauchy_isf"><a href="#val-cauchy_isf" class="anchor"></a><code><span class="keyword">val</span> cauchy_isf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_rvs"><a href="#val-t_rvs" class="anchor"></a><code><span class="keyword">val</span> t_rvs : <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_pdf"><a href="#val-t_pdf" class="anchor"></a><code><span class="keyword">val</span> t_pdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_logpdf"><a href="#val-t_logpdf" class="anchor"></a><code><span class="keyword">val</span> t_logpdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_cdf"><a href="#val-t_cdf" class="anchor"></a><code><span class="keyword">val</span> t_cdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_logcdf"><a href="#val-t_logcdf" class="anchor"></a><code><span class="keyword">val</span> t_logcdf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_ppf"><a href="#val-t_ppf" class="anchor"></a><code><span class="keyword">val</span> t_ppf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_sf"><a href="#val-t_sf" class="anchor"></a><code><span class="keyword">val</span> t_sf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_logsf"><a href="#val-t_logsf" class="anchor"></a><code><span class="keyword">val</span> t_logsf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-t_isf"><a href="#val-t_isf" class="anchor"></a><code><span class="keyword">val</span> t_isf : float <span>&#45;&gt;</span> <span>df:float</span> <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_rvs"><a href="#val-vonmises_rvs" class="anchor"></a><code><span class="keyword">val</span> vonmises_rvs : <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_pdf"><a href="#val-vonmises_pdf" class="anchor"></a><code><span class="keyword">val</span> vonmises_pdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_logpdf"><a href="#val-vonmises_logpdf" class="anchor"></a><code><span class="keyword">val</span> vonmises_logpdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_cdf"><a href="#val-vonmises_cdf" class="anchor"></a><code><span class="keyword">val</span> vonmises_cdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_logcdf"><a href="#val-vonmises_logcdf" class="anchor"></a><code><span class="keyword">val</span> vonmises_logcdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_sf"><a href="#val-vonmises_sf" class="anchor"></a><code><span class="keyword">val</span> vonmises_sf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-vonmises_logsf"><a href="#val-vonmises_logsf" class="anchor"></a><code><span class="keyword">val</span> vonmises_logsf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>kappa:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_rvs"><a href="#val-lomax_rvs" class="anchor"></a><code><span class="keyword">val</span> lomax_rvs : <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_pdf"><a href="#val-lomax_pdf" class="anchor"></a><code><span class="keyword">val</span> lomax_pdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_logpdf"><a href="#val-lomax_logpdf" class="anchor"></a><code><span class="keyword">val</span> lomax_logpdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_cdf"><a href="#val-lomax_cdf" class="anchor"></a><code><span class="keyword">val</span> lomax_cdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_logcdf"><a href="#val-lomax_logcdf" class="anchor"></a><code><span class="keyword">val</span> lomax_logcdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_ppf"><a href="#val-lomax_ppf" class="anchor"></a><code><span class="keyword">val</span> lomax_ppf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_sf"><a href="#val-lomax_sf" class="anchor"></a><code><span class="keyword">val</span> lomax_sf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_logsf"><a href="#val-lomax_logsf" class="anchor"></a><code><span class="keyword">val</span> lomax_logsf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lomax_isf"><a href="#val-lomax_isf" class="anchor"></a><code><span class="keyword">val</span> lomax_isf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_rvs"><a href="#val-weibull_rvs" class="anchor"></a><code><span class="keyword">val</span> weibull_rvs : <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_pdf"><a href="#val-weibull_pdf" class="anchor"></a><code><span class="keyword">val</span> weibull_pdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_logpdf"><a href="#val-weibull_logpdf" class="anchor"></a><code><span class="keyword">val</span> weibull_logpdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_cdf"><a href="#val-weibull_cdf" class="anchor"></a><code><span class="keyword">val</span> weibull_cdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_logcdf"><a href="#val-weibull_logcdf" class="anchor"></a><code><span class="keyword">val</span> weibull_logcdf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_ppf"><a href="#val-weibull_ppf" class="anchor"></a><code><span class="keyword">val</span> weibull_ppf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_sf"><a href="#val-weibull_sf" class="anchor"></a><code><span class="keyword">val</span> weibull_sf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_logsf"><a href="#val-weibull_logsf" class="anchor"></a><code><span class="keyword">val</span> weibull_logsf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-weibull_isf"><a href="#val-weibull_isf" class="anchor"></a><code><span class="keyword">val</span> weibull_isf : float <span>&#45;&gt;</span> <span>shape:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_rvs"><a href="#val-laplace_rvs" class="anchor"></a><code><span class="keyword">val</span> laplace_rvs : <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_pdf"><a href="#val-laplace_pdf" class="anchor"></a><code><span class="keyword">val</span> laplace_pdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_logpdf"><a href="#val-laplace_logpdf" class="anchor"></a><code><span class="keyword">val</span> laplace_logpdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_cdf"><a href="#val-laplace_cdf" class="anchor"></a><code><span class="keyword">val</span> laplace_cdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_logcdf"><a href="#val-laplace_logcdf" class="anchor"></a><code><span class="keyword">val</span> laplace_logcdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_ppf"><a href="#val-laplace_ppf" class="anchor"></a><code><span class="keyword">val</span> laplace_ppf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_sf"><a href="#val-laplace_sf" class="anchor"></a><code><span class="keyword">val</span> laplace_sf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_logsf"><a href="#val-laplace_logsf" class="anchor"></a><code><span class="keyword">val</span> laplace_logsf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-laplace_isf"><a href="#val-laplace_isf" class="anchor"></a><code><span class="keyword">val</span> laplace_isf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_rvs"><a href="#val-gumbel1_rvs" class="anchor"></a><code><span class="keyword">val</span> gumbel1_rvs : <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_pdf"><a href="#val-gumbel1_pdf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_pdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_logpdf"><a href="#val-gumbel1_logpdf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_logpdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_cdf"><a href="#val-gumbel1_cdf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_cdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_logcdf"><a href="#val-gumbel1_logcdf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_logcdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_ppf"><a href="#val-gumbel1_ppf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_ppf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_sf"><a href="#val-gumbel1_sf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_sf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_logsf"><a href="#val-gumbel1_logsf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_logsf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel1_isf"><a href="#val-gumbel1_isf" class="anchor"></a><code><span class="keyword">val</span> gumbel1_isf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_rvs"><a href="#val-gumbel2_rvs" class="anchor"></a><code><span class="keyword">val</span> gumbel2_rvs : <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_pdf"><a href="#val-gumbel2_pdf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_pdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_logpdf"><a href="#val-gumbel2_logpdf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_logpdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_cdf"><a href="#val-gumbel2_cdf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_cdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_logcdf"><a href="#val-gumbel2_logcdf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_logcdf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_ppf"><a href="#val-gumbel2_ppf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_ppf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_sf"><a href="#val-gumbel2_sf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_sf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_logsf"><a href="#val-gumbel2_logsf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_logsf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-gumbel2_isf"><a href="#val-gumbel2_isf" class="anchor"></a><code><span class="keyword">val</span> gumbel2_isf : float <span>&#45;&gt;</span> <span>a:float</span> <span>&#45;&gt;</span> <span>b:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_rvs"><a href="#val-logistic_rvs" class="anchor"></a><code><span class="keyword">val</span> logistic_rvs : <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_pdf"><a href="#val-logistic_pdf" class="anchor"></a><code><span class="keyword">val</span> logistic_pdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_logpdf"><a href="#val-logistic_logpdf" class="anchor"></a><code><span class="keyword">val</span> logistic_logpdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_cdf"><a href="#val-logistic_cdf" class="anchor"></a><code><span class="keyword">val</span> logistic_cdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_logcdf"><a href="#val-logistic_logcdf" class="anchor"></a><code><span class="keyword">val</span> logistic_logcdf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_ppf"><a href="#val-logistic_ppf" class="anchor"></a><code><span class="keyword">val</span> logistic_ppf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_sf"><a href="#val-logistic_sf" class="anchor"></a><code><span class="keyword">val</span> logistic_sf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_logsf"><a href="#val-logistic_logsf" class="anchor"></a><code><span class="keyword">val</span> logistic_logsf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-logistic_isf"><a href="#val-logistic_isf" class="anchor"></a><code><span class="keyword">val</span> logistic_isf : float <span>&#45;&gt;</span> <span>loc:float</span> <span>&#45;&gt;</span> <span>scale:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_rvs"><a href="#val-lognormal_rvs" class="anchor"></a><code><span class="keyword">val</span> lognormal_rvs : <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_pdf"><a href="#val-lognormal_pdf" class="anchor"></a><code><span class="keyword">val</span> lognormal_pdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_logpdf"><a href="#val-lognormal_logpdf" class="anchor"></a><code><span class="keyword">val</span> lognormal_logpdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_cdf"><a href="#val-lognormal_cdf" class="anchor"></a><code><span class="keyword">val</span> lognormal_cdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_logcdf"><a href="#val-lognormal_logcdf" class="anchor"></a><code><span class="keyword">val</span> lognormal_logcdf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_ppf"><a href="#val-lognormal_ppf" class="anchor"></a><code><span class="keyword">val</span> lognormal_ppf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_sf"><a href="#val-lognormal_sf" class="anchor"></a><code><span class="keyword">val</span> lognormal_sf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_logsf"><a href="#val-lognormal_logsf" class="anchor"></a><code><span class="keyword">val</span> lognormal_logsf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-lognormal_isf"><a href="#val-lognormal_isf" class="anchor"></a><code><span class="keyword">val</span> lognormal_isf : float <span>&#45;&gt;</span> <span>mu:float</span> <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_rvs"><a href="#val-rayleigh_rvs" class="anchor"></a><code><span class="keyword">val</span> rayleigh_rvs : <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_pdf"><a href="#val-rayleigh_pdf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_pdf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_logpdf"><a href="#val-rayleigh_logpdf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_logpdf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_cdf"><a href="#val-rayleigh_cdf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_cdf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_logcdf"><a href="#val-rayleigh_logcdf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_logcdf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_ppf"><a href="#val-rayleigh_ppf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_ppf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_sf"><a href="#val-rayleigh_sf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_sf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_logsf"><a href="#val-rayleigh_logsf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_logsf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-rayleigh_isf"><a href="#val-rayleigh_isf" class="anchor"></a><code><span class="keyword">val</span> rayleigh_isf : float <span>&#45;&gt;</span> <span>sigma:float</span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-dirichlet_rvs"><a href="#val-dirichlet_rvs" class="anchor"></a><code><span class="keyword">val</span> dirichlet_rvs : <span>alpha:<span>float array</span></span> <span>&#45;&gt;</span> <span>float array</span></code></dt><dd><p>``dirichlet_rvs ~alpha`` returns random variables of ``K-1`` order Dirichlet distribution, follows the following probabilty dense function.</p><p>.. math:: f(x_1,...,x_K; \alpha_1,...,\alpha_K) = \frac</p><h2 id=""><a href="#" class="anchor"></a></h2><p><code>\mathbf{B(\alpha)</code></p><p>}</p><p>\prod_<i>=1</i>^K x_i^<code>\alpha_i - 1</code></p><p>The normalising constant is the multivariate Beta function, which can be expressed in terms of the gamma function:</p><p>.. math:: \mathbf<code>B(\alpha)</code> = \frac<code>\prod_{i=1</code>^K \Gamma(\alpha_i)</p><p>}</p><p><code>\Gamma(\sum_{i=1</code>^K \alpha_i)</p><p>}</p><p>Note that ``x`` is a standard K-1 simplex, i.e. :math:`\sum_i^K x_i = 1` and :math:`x_i \ge 0, \forall x_i \in <code>1,K</code>`.</p></dd></dl><dl><dt class="spec value" id="val-dirichlet_pdf"><a href="#val-dirichlet_pdf" class="anchor"></a><code><span class="keyword">val</span> dirichlet_pdf : <span>float array</span> <span>&#45;&gt;</span> <span>alpha:<span>float array</span></span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl><dl><dt class="spec value" id="val-dirichlet_logpdf"><a href="#val-dirichlet_logpdf" class="anchor"></a><code><span class="keyword">val</span> dirichlet_logpdf : <span>float array</span> <span>&#45;&gt;</span> <span>alpha:<span>float array</span></span> <span>&#45;&gt;</span> float</code></dt><dd><p>TODO</p></dd></dl></section></div></body></html>